1. Increasing & Decreasing Functions & The First Derivative Test (3.3)
December 13th, 2011

2. I. Increasing & Decreasing Functions
Defs. of Increasing & Decreasing Functions: A function f is increasing on an interval if for any two numbers x1 and x2 in the interval x1< x2 implies f(x1)<f(x2).
A function f is decreasing on an interval if for any two numbers x1 and x2 in the interval x1< x2 implies f(x1)>f(x2).

3. *Increasing is going up from left to right, decreasing is going down from left to right.

4. Thm. 3.5: Test for Increasing & Decreasing Functions: Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b).
1. If f’(x)>0 for all x in (a, b), then f is increasing on [a, b], (because f has a positive slope on (a, b)).
2. If f’(x)<0 for all x in (a, b), then f is decreasing on [a, b], (because f has a negative slope on (a, b)).
3. If f’(x)=0 for all x in (a, b), then f is constant on
[a, b], (because f is a horizontal line & has a slope of 0 on (a, b)).

5. decreasing
f’(x)<0
increasing
f’(x)>0
constant
f’(x)=0

6. Ex. 1: Find the open intervals on which
is increasing or decreasing.

8. Guidelines for Finding Intervals on Which a Function is Increasing or Decreasing: Let f be continuous on the interval (a, b). To find the open intervals on which f is increasing or decreasing, use the following steps.
1. Locate the critical numbers of f in (a, b) and use these numbers to determine the test intervals.
2. Determine the sign of f’(x) at one test value in each interval.
3. Use Thm. 3.5 to determine whether f is increasing or decreasing on each interval.

9. You Try: Identify the open intervals on which the function is increasing or decreasing.

10. II. The First Derivative Test
Thm. 3.6: The First Derivative Test: Let c be a critical number of a function f that is continuous on the open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows.

11. 1. If f’(x) changes from negative to positive at c, then f has a relative minimum at (c, f(c)).
(-)
(+)
(c, f(c))

12. 2. If f’(x) changes from positive to negative at c, then f has a relative maximum at (c, f(c)).
(+)
(-)
(c, f(c))
f’(x)>0
f’(x)<0

13. 3. If f’(x) is positive on both sides of c or negative on both sides of c, then f(c) is neither a relative minimum nor a relative maximum.
(c, f(c))
(+)
(+)
f’(x)>0
f’(x)>0

14. Ex. 2: Find the relative extrema of the function on the interval .

15. You Try:Find the relative extrema of the function .